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The Significance of the Number 108

Readers familiar with Asian traditions will recognize the importance of the number 108. It holds deep meaning in Hinduism, Jainism, Sikhism, Buddhism, and Chinese culture. It appears in many contexts—rosary beads, yoga practices, symbolic carvings, and even emergency services.

Beyond its cultural importance, 108 has remarkable mathematical properties, as highlighted in a book written by Shyam Sunder Gupta, a former senior railway engineer. Here are some examples:

- 108 = 1¹ × 2² × 3³ (the product of the first three numbers each raised to themselves).
- It is the smallest number whose divisors include every digit at least once.
- It is the largest known number n where 2ⁿ does not contain the digit 9.
- It represents the internal angle (in degrees) of a regular pentagon.
- It is the number of free heptominoes (shapes made from seven connected squares).
- It also matches the count of cards in a UNO deck.

Gupta’s book, Exploring the Beauty of Fascinating Numbers, dedicates extensive discussion to 108, noting that its digits symbolize truth, emptiness, and infinity. A similar focus is given to pi, 13, 666, and 153, which may be the second most intriguing number.

The cultural reverence for 108 might have inspired mathematicians to explore its properties, or perhaps its mathematical uniqueness led to its spiritual significance. Either way, it is fascinating how numerology and mathematics converge on this subject.

### Puzzles on Notable Numbers

Today’s challenges revolve around numbers with unusual traits.

1. Counting Combinations
You have ten cards, each displaying a digit from 0 to 9. Arranged in order, they form numbers ranging from 0123456789 to 9876543210.

i) How many of these numbers are divisible by 2?
ii) How many are divisible by 3?

2. Smallest Even Four-Digit Number
What is the smallest even number between 1000 and 9999 that uses four different digits?

3. Self-Descriptive Number
An autobiographical number is defined such that:
- The first digit counts the number of 0s,
- The second counts the 1s,
- And so on.
For example, 1210 has 1 zero, 2 ones, 1 two, and 0 threes.

Find the only ten-digit autobiographical number.